Vis ComputDOI 10.1007/s00371-014-1041-3

ORIGINAL ARTICLE

Geodesic bifurcation on smooth surfaces

Hannes Thielhelm · Alexander Vais ·Franz-Erich Wolter

© Springer-Verlag Berlin Heidelberg 2014

Abstract Within Riemannian geometry the geodesic expo-nential map is an essential tool for various distance-relatedinvestigations and computations. Several natural questionscan be formulated in terms of its preimages, usually leadingto quite challenging non-linear problems. In this context werecently proposed an approach for computing multiple geo-desics connecting two arbitrary points on two-dimensionalsurfaces in situations where an ambiguity of these connectinggeodesics is indicated by the presence of focal curves. Theessence of the approach consists in exploiting the structureof the associated focal curve and using a suitable curve fora homotopy algorithm to collect the geodesic connections.In this follow-up discussion we extend those constructionsto overcome a significant limitation inherent in the previ-ous method, i.e. the necessity to construct homotopy curvesartificially. We show that considering homotopy curves meet-ing a focal curve tangentially leads to a singularity that weinvestigate thoroughly. Solving this so-called geodesic bifur-cation analytically and dealing with it numerically providesnot only theoretical insights, but also allows geodesics to beused as homotopy curves. This yields a stable computationaltool in the context of computing distances. This is applica-ble in common situations where there is a curvature inducednon-injectivity of the exponential map. In particular we illus-trate how applying geodesic bifurcation approaches the dis-tance problem on compact manifolds with a single closedfocal curve. Furthermore, the presented investigations pro-vide natural initial values for computing cut loci using themedial differential equation which directly leads to a discus-

H. Thielhelm (B) · A. Vais · F.-E. WolterWelfenlab, Leibniz University of Hannover, Hannover, Germanye-mail: thielhel@welfenlab.de

A. Vaise-mail: vais@welfenlab.de

sion on avoiding redundant computations by combining thepresented concepts to determine branching points.

Keywords Geodesic exponential map · Focal curves ·Connecting geodesics · Distance computation · Cut locus ·Voronoi diagram

1 Introduction

Distance-related problems including notions such as Voronoidiagrams or the cut locus are a central topic in computa-tional geometry, where they are considered in Euclidean andalso in more abstract settings, see e.g. [4]. Common dis-crete approaches dealing with non-Euclidean situations usu-ally base or implicitly rely on a suitable approximation of thedistance function dM (p, ·) with respect to some point p onthe considered object M , see e.g. [14,33]. The latter is com-monly embedded into Euclidean space and discretized ade-quately. For example, combinatorial graph based techniquessuch as the Dijkstra algorithm typically perform a sweepover M emanating from p and extend every path which isdistance minimizing. Even more sophisticated discrete dis-tance approximations such as fast marching methods [10]basically iteratively extend a frontier of shortest paths andcompare distances to preserve the minimizing property. Assoon as a path ceases to be a globally shortest one, it is notconsidered anymore within such approaches as it loses itsrelevance for the discrete distance approximation.

The theory on Riemannian manifolds draws a much richerpicture, incorporating curvature information to analyze thedistance problem on M . Most notably it is well-known thatshortest paths have the property of being locally straightestcurves, known as geodesics, which are generalizations of thestraight lines in the Euclidean setting. In the Riemannian case

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Fig. 1 Generic situation

not every geodesic has to be a shortest path. However, thereverse implication is true, therefore it is sufficient to searchfor shortest paths within the set of all geodesics connectingtwo given points p, q ∈ M . This is not possible within thecommonly used discrete frameworks, since they lack the con-cept of a geodesic exponential map. Note that [25] introducesan exponential map in polyhedral settings paving the way toexploit the concept of geodesics within discrete settings.

The non-injectivity of the latter is captured in the cut locus.More precisely the cut locus of a point p ∈ M is the clo-sure of all points in M where a geodesic starting in p loosesits distance minimizing property. Its relevance for the dis-tance problem arises from the fact, that any geodesic notintersecting the cut locus is a shortest path. Consider Fig. 1showing a surface patch and three geodesics connecting pand q. While the cyan geodesic is the shortest path, since itdoes not intersect the cut locus (light red), the blue geodesicslose their distance-minimizing property at their intersectionpoints with the cut locus. We refer to such a set of connect-ing geodesics between p and q as distance relevant, since itcontains the shortest path. Once such a set of geodesics isavailable determining the distance is essentially trivial as itamounts to choosing the shortest of them.

It is possible to obtain distance relevant sets of connectinggeodesics without the use of the global cut locus conceptrelying on the local concept of focal curves. This approachhas been pursued in [34], which we follow in this paper.

In Fig. 1 the focal curve of p, also known as the conjugatelocus of p, is shown in dark red. Note that the cut locusbegins in the so-called cusp of the focal curve. This situationis actually a generic representative of a curvature-inducedcut locus branch, thereby focal cusps are considered to benatural starting points for tracing the cut locus.

Theoretical investigations for the cut locus can be foundfor example in [22,28,36,37]. Computing cut loci and relatedconcepts such as the medial axis or Voronoi diagrams based

Fig. 2 Voronoi diagram and a medial axis. Proof of concepts in curved3-space from [20,21]

on discrete respectively smooth surface representations hasreceived some attention in the literature, see e.g. [5,6,9,16],respectively [30,31]. The two and three-dimensional smoothRiemannian setting has been considered in [11,19,26] and[38,39], the latter two being historical overviews on therespective computational methods. A proof of concept forthe feasibility of computing three-dimensional Voronoi dia-grams respectively medial axis inverse transformations inthis case has been described in [20,21], see Fig. 2 for exem-plary results. However the corresponding two- and three-dimensional algorithms have been restricted to domains inwhich the exponential map is assumed to be injective, guar-anteeing unique geodesic connections and thereby induc-ing situations that are topologically similar to the familiarEuclidean setting. For related theoretical investigations onhow these restrictions can be expressed in terms of curvaturedependent bounds see e.g. [13,15,23]. A survey on computa-tional methods dealing with smooth surfaces can be found in[24]. In order to overcome the significant assumption of geo-desic injectivity we recently started a more detailed investi-gation of the general two-dimensional Riemannian case [34].We especially focus on a homotopy approach that general-izes to higher dimensions without exponentially increasingcomplexity. In order to illuminate the applicability of thisapproach within higher-dimensional distance related prob-lems we present two-dimensional examples which should beunderstood as a proof of concept.

In this paper we extend upon the homotopy approach (HA)presented in [34] for computing multiple connecting geo-desics on curved smooth surfaces. The HA collects a set ofconnecting geodesics between two arbitrary points p and q,making use of local concepts such as Jacobi fields and focalcurves in a straight-forward manner as indicated by classicaltheory. Note that aside from distance computations connect-ing geodesics are of independent interest when consideringthem within a variational context as stationary points of acorresponding energy functional, see e.g. [1,12].

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The main subject of the HA is to exploit the generic sit-uation shown in Fig. 1 where the Gaussian curvature of asurface patch causes a family of geodesics radiating fromp to intersect among themselves. The resulting ambigu-ity of connecting geodesics is indicated by the presenceof a focal curve. We refer to this situation as a locallycaused non-injectivity of the geodesic exponential map.However, in general the ambiguity of connecting geodes-ics can also be caused by the topology of M , which werefer to as a global origin of non-injectivity. This is indi-cated by the existence of geodesic loops not generatingfocal points within M . A familiar example is the cylinderthat exhibits geodesic loops even though it is intrinsicallyflat.

The HA is able to capture locally caused non-injectivi-ties, indicating it to be a useful tool for distance compu-tations on curved surfaces. However, a crucial part of themethod is to construct a suitable homotopy curve, which hasto intersect the focal curve transversally. In case the homo-topy curve meets the focal curve tangentially the equationswithin the HA exhibit a singularity, the so-called geodesicbifurcation. This singularity excludes geodesics that gener-ate a focal point as candidates for homotopy curves withinthe HA.

In this paper we fill this gap by studying and describinghow to deal with the geodesic bifurcation. It turns out theexamined singularity can be resolved enabling us to use geo-desics as canonical homotopy curves within the HA. Sincewe base our approach on the ability to evaluate the geodesicexponential map, geodesics are a priori easily constructedhomotopy curves in our context.

Additionally in a setting with closed focal curves ourpresent contribution allows to compute the distance of arbi-trary points without requiring the explicit knowledge of thecut locus or the focal curve. We exemplify how this solvesthe corresponding global distance problem on ellipsoidalshapes.

Our investigations of the geodesic bifurcation in focalcusps allow us to explain the singularity of the medial dif-ferential equation arising at the endpoints of the cut locus.Based on that we present a solution on how to deal withthis singularity in practical applications. Within this dis-cussion we explain the behaviour of medial curves in thepresence of focal curves and furthermore combine the pre-sented concepts to yield natural starting points for a redun-dancy minimizing approach to determine cut loci of dis-crete point sets, also commonly known as Voronoi dia-grams.

Our approach is applicable to parametrized or implicit sur-faces that allow the evaluation of second order derivatives,i.e. curvature information. This covers for example NURBSpatches or subdivision surfaces [32], as well as manifoldbased constructions.

2 Basics

In this section we introduce some tools of Riemannian geom-etry, necessarily omitting some technicalities and details dueto the lack of space. We follow the definitions and notations of[34] and recommend [3] for a detailed exposition to classicaldifferential geometry that we build upon.

2.1 Riemannian manifolds

In this paper we assume M to be a two-dimensional completeRiemannian manifold with metric tensor g = 〈·, ·〉. The latteris used to define the length of curves and induces a classicalmetric dM on M . In many applications M is assumed to bea smooth surface embedded in Euclidean 3-space inheritingthe ambient metric. However, our approach is designed toalso cover the intrinsic setting which is naturally related toenergy minimization problems as indicated by Maupertuis’principle, see [12].

The theorem of Hopf–Rinow assures the existence ofshortest paths, which are curves realizing the distancebetween two arbitrary points p, q ∈ M . We denote the tan-gent space of M in p by Tp M and the vector resulting froma positive quarter turn of a tangent vector v ∈ Tp M by v⊥.Since the following considerations take place within a localcontext we do not distinguish between the objects on themanifold and their representation within a particular chartx : U ⊂ M → R

2. The latter map identifies points p ∈ Mwith their coordinates (x1(p), x2(p)), simply denoted by(p1, p2). In this paper assume p ∈ M to be arbitrary, butfixed. Derivatives will be denoted as dotted quantities forfunctions of a single variable. We will often use the short-hand notation hu for the partial derivative ∂h

∂u .In the next sections we will briefly discuss how to evaluate

the geodesic exponential map, followed by a short review ofconcepts such as Jacobi fields, focal curves and the cut locus.

2.2 Geodesic polar coordinates (GPCs)

Geodesics on a Riemannian manifold can be understoodas generalizations of the straight lines in Euclidean space.In order to investigate geodesics emanating from a givenpoint p one usually introduces the geodesic exponential map,denoted by expp : Tp M → M , mapping s · v(ϕ) ∈ Tp Mto the endpoint of the geodesic γϕ starting in p in the direc-tion v(ϕ) ∈ S1 ⊂ Tp M with length s. The computation ofthe geodesic γ = γϕ : [0, s] → M is realized by solvingthe geodesic differential equations ∇γ γ = 0, where ∇ is theLevi–Civita connection on M . To evaluate expp in practiceone has to compute the coordinates of γ . This amounts tosolving

γ k + �ki j γ

i γ j = 0

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starting in a chart covering p with the initial values γ l(0) =pl , γ l(0) = vl and performing chart transitions during theintegration if necessary. The �k

i j are the Christoffel symbolsof ∇ and given by

�ki j = 1

2gmk

(∂gim

∂x j+ ∂g jm

∂xi− ∂gi j

∂xm

),

where (gi j ) is the representation of the metric tensor g withrespect to the frame induced by the current chart and (g jk)

denotes its inverse.Since the geodesics we use are by definition arc-length

parametrized, the parameters (s, ϕ) introduced by the mapOp : (s, ϕ) → γϕ(s) are said to be geodesic polar coordi-nates (GPCs) with respect to p on M . Note that Op is notinjective in general for s ≥ rp, where rp is chosen minimallyand said to be the injectivity radius of M with respect to p. Incase rp is finite Op does not equip M with proper coordinatesin a unique manner. The theorem of Hopf–Rinow howeverstates that Op is surjective, so that for any point q in M thereexists at least some s0, ϕ0 with Op(s0, ϕ0) = q. The cor-responding geodesic γϕ0 is called a connecting geodesic ofp = γ (0) and q = γ (s0).

The GPC concept allows us to define a geodesic circlewith center p ∈ M and radius s > 0 as being the set ofall points {Op(s, ϕ) : ϕ ∈ S1} which is a superset of thedistance circle {q ∈ M : dM (p, q) = s}.

We will now discuss the injectivity of Op incorporating theclassical concepts of focal curves respectively the cut locus.

2.3 Focal curves

To investigate the local injectivity of Op in terms of focalcurves we have to consider the vector field Jϕ0(s) =∂∂ϕ

Op(s, ϕ0) along the geodesic γ = γϕ0 for a fixed ϕ0.From the Lemma of Gauss it is evident, that Jϕ0(s) ⊥ γϕ0(s)and therefore Jϕ0(s) = y(s, ϕ0)γ

⊥ϕ0(s) respectively

∂

∂ϕOp(s, ϕ0) = y(s, ϕ)

∂

∂sOp(s, ϕ0)

⊥ (1)

with a scalar function y(·) = y(·, ϕ0). It is a well-knownresult that y satisfies the scalar Jacobi equation

y(s)+ K (γ (s))y(s) = 0 ,

where K (γ (s)) is the Gaussian curvature of M in γ (s). Inpractice the computation of y is realized by solving the abovedifferential equation simultaneously to the geodesic differ-ential equation for γ . Initial values for the considered Jacobifield are given by y(0) = 0 respectively y(0) = 1. See [34]for a derivation of the scalar Jacobi equation from its clas-sical formulation. Roughly spoken the spreading of infini-tesimally nearby geodesics in our two-dimensional setting isencoded into y and is closely related to Gaussian curvatureas described by the Jacobi equation.

Fig. 3 Closed respectively unbounded focal curves

A point a = Op(s0, ϕ0) is said to be conjugate to p ify(s0, ϕ0) = 0. In general geodesics can generate multiplefocal points while they extend, but since we use focal pointsas an indicator for non-injectivity we will direct our attentionto the first ones. The set of all these points is called the (first)conjugate locus. A generalization of the conjugate locus of pincorporating the Fermi coordinates with respect to a curvec on M instead of GPCs leads to the notion of focal curves[3]. Since the HA generalizes to yield connecting geodesicsemanating orthogonally from c, we will use the name focalcurve instead of conjugate locus. Thus, we say that the geo-desic γϕ0 generates the focal point a, and denote by f p theset of all focal points with respect to p.

If Op(s0, ϕ0) is a focal point then Op is not injective inany neighborhood of (s0, ϕ0) since the differential

DOp =(∂Op

∂s

∂Op

∂ϕ

)=

(γϕ0 yγ⊥

ϕ0

)

is singular as y(s0, ϕ0) vanishes [29]. In fact f p can be under-stood as bordering a region of local non-injectivity of theexponential map.

The success of finding all relevant geodesic connectionsfrom p within the HA depends on the specific choice ofthe homotopy curve and its interaction with f p. In order toconstruct suitable homotopy curves it is useful to distinguishtwo different types of focal curves.

The focal radius s f : S1 → R∪ {∞} is implicitly definedby y(s f (ϕ), ϕ) = 0, where s f (ϕ) is chosen to be minimal(thereby capturing the distance to the first occurrence of afocal point) or set to ∞ if γϕ does not generate a focal point.Let D ⊆ S1 be the set, where s f is finite. A focal curve issaid to be closed if D = S1, as for example on the ellipsoid inFig. 3. Otherwise, D consists of disjoint intervals of S1 andwe say that every connected component of f p correspondingto such an interval is unbounded. We will by abuse of lan-guage also refer to one component of f p as a focal curve ofp. See Fig. 3 on the right for a couple of unbounded focalcurves on a height surface.

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Consider now a focal curve of p, parametrized by f p(ϕ) =Op(s f (ϕ), ϕ). Differentiating yields

f p(ϕ) = s f (ϕ)∂Op

∂s(s f (ϕ), ϕ) = s f (ϕ)γϕ(s f (ϕ)) , (2)

where γϕ generates the focal point f p(ϕ). We say that f p(ϕ)

is a regular focal point if f p(ϕ) �= 0 and conclude from thelast equation that the geodesic γϕ generating the focal pointf p(ϕ)meets f p tangentially at that point. Since f p vanishesat points where s f becomes extremal we refer to such focalpoints as focal cusps. Differentiating y(s f (ϕ), ϕ) = 0 oneobtains

s f (ϕ) = − yϕ(s f (ϕ), ϕ)

ys(s f (ϕ), ϕ),

where the denominator never vanishes as discussed in[34]. A focal cusp Op(s0, ϕ0) is therefore characterized byyϕ(s0, ϕ0) = 0 in addition to y(s0, ϕ0) = 0.

2.4 Cut locus

Regarding the global injectivity of Op one usually considersthe concept of the cut locus [3,36]. An important differencebetween the cut locus C p and the focal curve f p of a pointp ∈ M is that the latter is defined in terms of locally availableinformation, i.e. it requires only knowledge of the metrictensor along the geodesic generating the considered focalpoint. The cut locus however is defined in a global sense,since it incorporates the global notion of a shortest path.

Let sc(ϕ) ∈ R∪{∞} denote the minimal parameter valuesuch that the geodesic s → γϕ0(s) stops being a shortest pathfor s > sc(ϕ). It is useful to set sc(ϕ) = ∞, if the geodesicγϕ remains a shortest path while it extends infinitely. Thenthe cut locus of p is given by

C p = cl{

Op(sc(ϕ), ϕ) |ϕ ∈ S1, sc(ϕ) < ∞},

where cl denotes the topological closure. It is well-knownthat rp ≤ sc(ϕ) ≤ s f (ϕ), i.e. the exponential map loosesits global injectivity in general before this becomes locallynoticeable.

A focal point f p(ϕ)where s f (ϕ) = sc(ϕ) has to be a focalcusp that is locally accessible as indicated by yϕ(s f (ϕ), ϕ) =0. Furthermore it lies on the topological boundary of C p.

In order to generalize our discussion, it is appropriateto consider a finite discrete set of reference points P ={p1, . . . , pn} instead of just a single point p. In order todistinguish between the reference points and other pointson M it will be convenient to also refer to the elements ofP as sites. We define the cut locus CP of P as the clo-sure of the set of all points having at least two shortestpaths to elements of P . This definition is compatible withthe above cut locus definition for a single reference point,[35]. Furthermore it generalizes the typical notion of the

Fig. 4 Symmetry set and cut locus

Voronoi diagram VP of P , the latter being usually definedin terms of a distance partition of M into Voronoi regionsR(p) = {q ∈ M : d(p, q) ≤ d(r, q) ∀r ∈ P}, whoseboundaries ∪p∈P∂R(p) form VP . It can be shown that ingeneral VP ⊂ CP , whereas in the Euclidean setting bothnotions coincide. Note that while the cut locus concept iseven more general as it allows for an arbitrary closed subsetof M to be used as a reference set, we will focus in this paperon the definition given above, conveniently denoting CP alsoas the (geodesic) Voronoi diagram of P .

Its local counterpart is called the symmetry set SP of P ={p1, . . . , pn} as given by

SP = cl{m| m = Opi (s, ϕ) = Op j (s, ψ) , (i, ϕ) �= ( j, ψ)

}i.e. as the closure of all points m ∈ M , which are connectedwith two distinct points pi , p j ∈ P via (at least) two geo-desics of equal length. The cut locus CP is contained in thesymmetry set SP , see e.g. Fig. 4, which by construction canbe determined using the medial equation

Op(s(λ), ϕ(λ)) = Oq(s(λ), ψ(λ)). (3)

By differentiating this equation with respect to λ one obtainsthe so-called medial differential equation (MDE, [27]), whichwill be used later on to determine geodesic Voronoi diagrams.

2.5 Structure of geodesic Voronoi diagrams

In the following we discuss the structure of geodesic Voronoidiagrams on a smooth surface which in the two-dimensionalcase is known to be a graph, c.f. [17,18], consisting of mul-tiple branches. In contrast to the cut locus consisting ofbranches we say that the symmetry set consists of medi-als, the latter being described by the medial equation. Asexplained above each branch of the cut locus is containedin the corresponding medial. We therefore first discuss thestructure of the medials and afterwards focus on the branch-ing behaviour of the cut locus, characterizing the latter interms of circumcircles.

For our purposes it is convenient to distinguish betweenlocally and globally induced medials with the correspond-ing cut locus branches being named analogously. More pre-cisely we will call a medial beginning respectively ending in

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Fig. 5 Two families of geodesics tracing a cut locus branch.

a focal cusp locally induced, whereas it is said to be globallyinduced otherwise. According to this definition any medialarising from considering the medial equation with two dis-tinct reference points is globally induced.

As an example for a locally induced medial consider Fig.5. The light red medial is traced by two families of geodesics(blue) starting in p as described by the medial differentialequation [with q = p in Eq. (3)]. Those families degenerateto a single geodesic (black) connecting p and the focal cusp,where the MDE becomes singular. However, the endpoint ofthe black geodesic is a point on the medial, which is locallyaccessible, as it is characterized as a focal cusp. The splittingof the single black geodesics into a pair of blue geodesicsis closely related to the geodesic bifurcation of the geodesicgenerating the focal cusp, see Sect. 3.3, and discussed in thecontext of natural starting points in Sect. 4.3.

Examples for globally induced medials are bisectorswithin a Voronoi diagram or the cut locus of a single ref-erence point on an unbounded cylinder. Notice that in bothcases one obtains a point on the medial as soon as one knowsa distance minimizing connecting geodesic [34] by using themiddle point on the corresponding geodesic.

Having briefly discussed medials we will now outline howthe intersections of medials give rise to the branching pointsof the cut locus. For this consider the Voronoi diagram forthe five sites (colored red) shown in Fig. 6 on the right. It istrivially contained in the respective symmetry set shown onthe left, which consists of several medials. In case the metricof the ambient surface is not Euclidean each of those medi-als is traced via the MDE incorporating appropriate initialpoints, as discussed later in Sect. 4.3. However, proceedingin this way one has to compute an initial point and a medialfor each pair of sites, resulting in redundant computations.This redundancy becomes evident in our example, where fivesites induce ten different medials (left) of which the dottedsegments (middle) are unnecessary, producing the final result(right).

In order to characterize the branching points of a geodesicVoronoi diagram CP , note that every branching point b in CP

has at least three distinct shortest geodesic connections to thesites in P , thereby determining a distance circle containingno sites in its interior. We will call such a circle circumcircleof the respective sites and its center b circumcenter. Our fivesite example contains four such relevant circumcircles, withone of those being depicted on the right of Fig. 6.

Note that contrary to the Euclidean case the consideredconnecting geodesics of a circumcircle can end up on thesame site. Furthermore, the circumcenters respectively cir-cumcircles do not need to be uniquely determined by threegiven sites. In fact the presented Riemannian generalizationsof the familiar Euclidean concepts exhibit many non-trivialphenomena, which we encounter in applications as witnessedin Sect. 4. This necessitates the tools developed and presentedin the following sections.

2.6 Derivatives

In this paper we will frequently deal with medials andhomotopy curves, being curves on M described by GPCsvia λ → Op(s(λ), ϕ(λ)). The tangent of such a curve is

directly given within the frame ∂Op∂s ,

∂Op∂s

⊥. The vector ∂Op

∂sis directly obtained from the geodesic differential equation

and ∂Op∂ϕ

= y∂Op∂s

⊥is calculated by solving the Jacobi equa-

tion for y.

Fig. 6 Redundant segments ofmedials

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The higher order derivatives of Op can be expressed viathe partial derivatives of y, which is calculated by solving thedifferentiated Jacobi equation discussed in [34]. First of all∇∂s∂Op∂s = ∇γ γ = 0 by the definition of a geodesic, implying

∇∂s∂Op∂s

⊥ = 0. Since ∂Op∂s and ∂Op

∂ϕare the basis fields induced

by geodesic polar coordinates, their Lie bracket vanishes.Taking into account that the Levi–Civita connection ∇ istorsion-free [3], this yields

∇∂ϕ

∂Op

∂s= ∇∂s

∂Op

∂ϕ,

For the derivatives with respect to the family parameter λweobtain the useful expressions

∇dλ

∂Op

∂s= ∇∂s

∂Op

∂ss + ∇

∂ϕ

∂Op

∂sϕ = ∇

∂s

∂Oq

∂ϕϕ

= ∇∂s

(y∂Op

∂s

⊥)ϕ = ys ϕ

∂Op

∂s

⊥,

∇dλ

∂Op

∂s

⊥= −ys ϕ

∂Op

∂s.

3 Contribution

In this section we extend the homotopy approach for com-puting connecting geodesics described in [34]. We beginby recalling this method briefly and motivate the geodesicbifurcation, which is investigated and discussed afterwards.Building on these results we describe how to use the geodesicbifurcation within the homotopy approach.

In general, surfaces can have a complicated distributionof positive and negative Gaussian curvature. To illustrateour approach we have included several pictures based onnumerical examples calculated using an implementation ofthe presented methods, focusing on the surface parametrizedby f (u, v) = (u, v, cos(u) + sin(v)), if not stated other-wise. While our methods deal with arbitrarily parametrizedsurfaces, possibly compact and covered with many charts,we have chosen this didactic example with regard to its richpresence of focal curves and the abundantly occurring asso-ciated phenomena studied in this paper. Please also note thatthe appearing boundaries are in fact a necessity of the visu-alization as all considered manifolds have no boundary.

3.1 Homotopy approach (HA)

For a brief illustration of the method we focus on the genericsituation shown in Fig. 7, where the curvature of a surfacepatch causes the geodesics emanating from p to intersect, asindicated by the presence of a focal curve f p, colored in darkred. We have placed a point q within the region bordered by

Fig. 7 Generic situation for the HA

the focal curve, which we will call the focal region withinthis example.

In order to calculate the connecting geodesics between pand q, [34] suggests to construct a regular curve Q : I → M(black) passing through q. This so-called homotopy curveQ can be chosen arbitrarily to a large extent, except that ithas to intersect the focal curve transversally and should startand end outside the focal region. Furthermore the homotopycurve is constructed to yield a geodesic connecting p withits start point Op(sb, ϕb) = Q(tb) = qb. A homotopy curvesatisfying all these properties will be called valid (within theHA). The HA allows for tracing a component of the one-dimensional solution set S of Op(s, ϕ) = Q(t), yielding asolution curve S : λ → (s(λ), ϕ(λ), t (λ)).

The solution set S contains information about the connect-ing geodesics between every point on the curve Q and p inthis setting. More precisely, if q = Q(tq), the initial valuesof the corresponding connecting geodesics are given by theGPCs of q:{(sq , ϕq) | (sq , ϕq , tq) ∈ S

}.

We refer to the process described above of determining thesolution curve S(λ) within the HA by saying that the curveQ is traced by the family of geodesics γϕ(λ) = Op(·, ϕ(λ)).

Figure 8 depicts the projection of the solution curve intothe (t, ϕ) parameter plane and illustrates how to collect thethree (blue) geodesic connections from p to q by consideringthat q = Q(tq). Within our prototypical setting and exempli-fied by Fig. 8, it is obvious that every point lying on Q andcontained in the interior of the focal region has exactly threeconnecting geodesics (blue) to p, while those points on theborder have two (cyan). The connecting geodesics from the

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Fig. 8 Solution curve of the HA in (t, ϕ)-space

points on Q, which are outside the focal region are unique(green). The characteristic behavior of the (t, ϕ)-solution inits extremal points is phrased by saying that the family γϕ(λ)is reflected by the focal curve, i.e. it reflects at points wherethe corresponding geodesic generates a focal point, markeddark red. For a more detailed explanation of the HA see [34].

3.2 Geodesic bifurcation

In order to explain and understand the geodesic bifurcationwe have to examine the solution curves obtained by tracinga collection of homotopy curves. For this purpose considerthe left part of Fig. 9 depicting three different homotopycurves starting in qb. The right part shows projections of thecorresponding solution curves in the (s, ϕ) parameter plane.It also visualizes the focal curve f p via s f (ϕ) in dark redappearing as a kind of parabola opening in s-direction withits lower part corresponding to the left arc of f p. The bluecurve on the left is a valid homotopy curve as it starts and endsoutside of the focal region bordered by f p while crossingf p transversally and the same applies for the green curve,although it is closer to being tangential to the left arc of thefocal curve. The cyan curve misses the left arc of the focalcurve completely and does not end outside the focal region,i.e. it is not valid.

The corresponding solution curves in the parameter dia-gram for the valid homotopy curves consist of one connectedcomponent. However, the cyan solution set consists of twoconnected components. The HA would only trace the upper

component of it, while the lower remains locally inaccessi-ble.

Imagine the three cases shown to be embedded within aone-parameter family of homotopy curves and correspond-ing configurations in the (s, ϕ)-plane. In this family we canexpect one homotopy curve to meet the focal curve tangen-tially. Note that the HA as presented in [34] fails for suchhomotopy curves, i.e. excluding geodesics generating a focalpoint from being used as homotopy curves. In the followingwe will explicitly deal with this tangential situation.

The solution curve S : λ → (s(λ), ϕ(λ), t (λ)) within theHA is characterized by

Op(s(λ), ϕ(λ)) = Q(t (λ)). (4)

Differentiating with respect to λ and using (1) yields

s∂Op

∂s+ yϕ

∂Op

∂s

⊥= t Q, (5)

In order to determine the tangent vector S(λ), considertaking scalar products to obtain

s =⟨

Q,∂Op

∂s

⟩t yϕ =

⟨Q,∂Op

∂s

⊥⟩t . (6)

Without loss of generality we assume Q to be parametri-zed by arc length. Then the above equations, together withthe additional condition s2 + ϕ2 + t2 = 1, can be solvedfor S(λ) = (s, ϕ, t) uniquely up to orientation, provided

that y and

⟨Q,

∂Op∂s

⊥⟩do not vanish simultaneously. This

tangent information can be used within classical numericalODE solvers or predictor-corrector methods as discussed in[34].

Consider now a point S(λ0) = (s0, ϕ0, t0) on the solutioncurve, where the above equations become singular and cannotbe uniquely solved, i.e. we have

y(s0, ϕ0) = 0 ⇔ Q(t0) ∈ f p ,⟨Q(t0),

∂Op∂s (s0, ϕ0)

⊥⟩= 0 ⇔ Q(t0) ‖ γϕ0 .

Using Eq. (2) we conclude that Q has to meet the focal curvef p tangentially in Q(t0) in order for the Eq. (6) to becomesingular and vice versa. In this case, we still obtain s0 = t0,

Fig. 9 Two valid curves(green, blue) and one non-validhomotopy curve (cyan) and theirpreimages under Op

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but cannot infer ϕ0 from (6) indicating that the solution setof (4) does not necessarily have the topology of a curve. Infact, it will turn out that it consists of two branches meetingin (s0, ϕ0, t0).

In the following we use the shorthand notation s0 =s(λ0), ϕ0 = ϕ(λ0), etc. The tangential vectors (s0, ϕ0) ofthose branches in the intersection point (s0, ϕ0) cannot beobtained from the first-order derivatives. Instead one mayapply L’Hôpital or equivalently differentiate Eq. (5) againwith respect to λ yielding

s∂Op

∂s+ s

∇∂λ

∂Op

∂s+ (ys ϕ + yϕ)

∂Op

∂s

⊥+ yϕ

∇∂λ

∂Op

∂s

⊥

= t∇∂λ

Q + t Q,

where the partial derivatives of Op respectively y are evalu-ated in (s0, ϕ0). Using the derivatives introduced in Sect. 2.6and collecting terms, one obtains

(s − yys ϕ

2) ∂Op

∂s+

(yϕ + 2ys sϕ + yϕϕ

2)∂Op

∂s

⊥

= t∇∂λ

Q + t Q. (7)

Substituting y(s0, ϕ0) = 0 and using the geodesic curva-ture κ of Q to express

∇∂λ

Q(t0) = κ(t0)Q⊥(t0) = κ(t0)

∂Op

∂s

⊥(s0, ϕ0),

equation (7) simplifies to

s0∂Op

∂s+

(yϕϕ

20 + 2ys s0ϕ0

) ∂Op

∂s

⊥

= t0∂Op

∂s+ t0κ(t0)

∂Op

∂s

⊥.

By comparing coefficients we obtain s0 = t0 and a quadraticequation for ϕ0:

yϕϕ20 + 2ys s0ϕ0 − t0κ(t0) = 0.

At this point, the presence of the two possible values

ϕ0 =−ys s0 ±

√y2

s s20 + t0κ(t0)yϕ

yϕ(8)

analytically illuminates the existence of two solution branchesmeeting in (s0, ϕ0, t0). Therefore, we speak of a geodesicbifurcation occurring.

The projection of the solution set S into the (s, ϕ)-planeis colored in black in the example shown in Fig. 12 and con-sists of two differentiable solution branches meeting in thebifurcation point (s0, ϕ0). These become accessible using thetangent information from Eq. (8) together with s0 = t0 fromEq. (6).

Using this tangent information adequately within the HA,as described in Sect. 3.4, the method yields two families ofgeodesics tracing Q, which correspond to the two solutionbranches. Therefore we consider curves, which meet the focalcurve tangentially, and especially geodesics generating focalpoints from now on as valid homotopy curves.

3.3 Geodesics as homotopy curves

From now on consider the homotopy curve Q to be a geodesicγϕ0 , which generates the focal point Op(s0, ϕ0). As Q is ageodesic, we haveκ(t0) = 0 in Eq. (8). In this case the tangentdirections of the two solution branches in the bifurcationpoint are given by

s0 = t0 and

(ϕ0 = 0 or ϕ0 = −2t0

ys

yϕ

). (9)

These branches are illustrated in Fig. 10, fitting into thecollection of solution sets depicted in Fig. 9. The blackstraight line ϕ = ϕ0 in the parameter diagram correspondstrivially to the (black) geodesic γϕ0 . The curved branch col-ored in black in the parameter diagram is the one we areinterested in. It becomes accessible using the non-trivial tan-gent information from the last equation.

Since the equations for the tangent information in thebifurcation/focal point require yϕ(s0, ϕ0) not to vanish, theycannot be used when Op(s0, ϕ0) is a focal cusp. We consider

Fig. 10 Geodesic (black) ashomotopy curve and its solutionbranches within the HA in black

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now this remaining case. The derived equations are not suf-ficient to obtain the tangent information in this very specialcase, which requires differentiating Eq. (7). Despite the com-pact presentation using the notational shortcuts outlined inSect. 2.6, it turns out that these calculations are quite tedious,though straightforward. Performing them yields besides s0 =0:

s0 = t0, ϕ0 = 0 or ϕ0 = ±√

−3t0ys

yϕϕ. (10)

In addition the same calculations can be used to determinethe higher order derivative ϕ0 in every regular bifurcationpoint: Here we already have s0 = t0 and obtain

ϕ0 = 4ysϕys

y2ϕ

− 8

3yϕϕ

y2s

y3ϕ

. (11)

Note that the presented equations in this section for the tan-gent or higher order information are only valid in the bifurca-tion point (s0, ϕ0, t0). However, they can obviously be usedto obtain first respectively second order Taylor approxima-tions of the solution branches, as shown in green respectivelyblue in Fig. 13 on the left. We have used a second respec-tively first order Taylor approximation for s respectively ϕin the focal cusp to yield the green parabola approximatingthe black solution branch in the figure on the right.

3.4 Geodesic bifurcation within the HA

Up to now we were mainly interested in the bifurcation phe-nomenon to resolve the singularity of the exponential mapin a focal point. In the following we combine the equationsfor the tangent information in non-singular points (6) and inthe bifurcation points (9) in order to use geodesics, whichgenerate a focal point, as homotopy curves within the HA.

Again, the homotopy curve Q is a geodesic γϕ0 generatinga focal point at Op(s0, ϕ0). The point (s0, ϕ0, t0)with t0 = s0

is the bifurcation point in the parameter space, i.e. it lies onthe non-trivial solution branch and serves as an initial valueto start tracing it. At this point, one has to use the tangentinformation from Eq. (9) to perform an initial step away fromthe singular bifurcation point and the trivial solution branch

Fig. 12 Two solution branches meeting in (s0, ϕ0)

ϕ = ϕ0. This can be achieved using a small Euler integrationstep based on the first order Taylor approximation using asmall step t = λ:

ϕ1 = ϕ0 − 2ys

yϕλ, s1 = s0 +λ, t1 = t0 +λ

The sign of t = λ actually determines the tracing directionon the solution path. A more accurate initial step is obtainedby using the second order information given by Eq. (11).Now the point (s1, ϕ1, t1) serves as an initial value suitablefor obtaining a family of geodesics tracing Q as describedby Eq. (6).

From now on we will refer to performing the HA with ageodesic γϕ0 (which generates a focal point) as homotopycurve, simply as applying geodesic bifurcation to the geo-desic γϕ0 . This process is illustrated in Fig. 11 where the geo-desic γϕ0 is colored black. The green geodesics are obtainedby tracing γϕ0 upwards. The purple geodesics trace it down-

Fig. 11 Family of geodesics tracing the black geodesic obtained by applying geodesic bifurcation

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Fig. 13 Left First (green) andsecond (blue) order Taylorapproximation of the solutionbranch (black) in a regular focalpoint. Right Mixed Taylorapproximation (green) of thesolution branch (black) in afocal cusp

wards, whereas after reflection tracing continues upwardsalong γϕ0 yielding the geodesics colored orange.

Note that aside from the tangent information, it is benefi-cial to exploit the original Eq. (4) to perform corrector stepsthat ensure one stays on the solution curve. These predictor-corrector methods [2,7] trace regular solution curves in gen-eral more efficiently than classical ODE solvers. However asour solution set fails to be a regular curve in the bifurcationpoint, one needs to be aware that after an inaccurate integra-tion step followed by corrector steps one may end up on thewrong branch. To ensure the accuracy of the initial step werecommend using the second-order information (11).

Observe that we cannot obtain the second order informa-tion ϕ0 in a focal cusp from the presented equations. How-ever, the angle between the two branches there is actuallyπ2 , making the initial integration step unproblematic. This iseasily seen by comparing the tangents of the two branchesand also visible in the numerical example depicted in Fig. 13on the right.

Aside from the numerical examples presented in thispapers, we have tested our approach not only with geodesicsas homotopy curves but also in case Q is a regular non-geodesic curve tangentially meeting the focal curve. In allcases the method exhibited a numerically stable behavior.Thus, we consider it to be applicable as an efficient compu-tational tool.

4 Applications

In this section we give a proof of concept for the geodesicbifurcation in order to outline its ability to capture distancerelated phenomena in various contexts. We strive for a pre-sentation which illuminates the structure of ambiguity of con-necting geodesics and how geodesic bifurcation is applied inthis context. As an introductory example we illustrate howthe HA profits in situations where geodesics present them-selves as natural homotopy curves. Afterwards, consideringexamples, we discuss how to use geodesic bifurcation forcomputing distances on manifolds having a single closedfocal curve.

Fig. 14 Focal curve f p and its preimage under Op

We also discuss how to apply the geodesic bifurcation tosolve the singularity of the medial differential equation in thefocal cusp, providing starting points for tracing the locallyinduced cut locus branches. Furthermore we combine themedial equation and the HA in order to determine circumcir-cles as branching points of geodesic Voronoi diagrams. Theseconsiderations lead to a computational approach avoidingredundant tracing of medial segments by exploiting theexamined natural starting points.

4.1 Introductory example

Figure 14 shows a generic situation on a height surface wherea focal curve f p is depicted in red. We consider the problemof determining the shortest paths from p to the focal curvef p. This problem reduces to the computation of all connect-ing geodesics from p to points on f p, i.e. to determining thepreimage of f p under Op. We use the blue geodesic γ toapply geodesic bifurcation in the focal point q, colored yel-

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Fig. 15 Preimages of the focal curve under expp

low. The solution branch of the geodesic bifurcation is shownin black in the parameter diagram. We follow this branch untilreaching the cyan point, yielding the cyan geodesic.

In order to use the curve f p(ϕ) as a homotopy curve withinthe HA, observe that the coordinates of the cyan-coloredpoint serve as suitable initial values. Thereby we obtain afamily of geodesics tracing the focal curve, consisting of theshortest paths from p to f p, represented by the cyan curve inthe (s, ϕ)-plane.

In principle the classical HA could also yield the cyangeodesic. However, this would require the construction ofa valid homotopy curve Q, i.e. one has to ensure that Qstarts and ends outside of the region bounded by f p, does notintersect f p tangentially and passes through the endpoint ofthe blue geodesic. Furthermore one would have to providesuitable geodesic polar coordinates of some point on Q asinitial values for tracing it. From an application point of viewthis is unsatisfying. However, note that in our context the bluegeodesic is a priori given and thereby suggests itself to applygeodesic bifurcation.

Figure 15 shows an analogous situation on an ellipsoid.Applying geodesic bifurcation to the blue geodesic yieldstwo other geodesics connecting p with the yellow focal point.Their GPCs serve as initial values for obtaining two familiesof geodesics tracing the focal curve. Again the cyan curve inthe parameter diagram consists of the GPCs of the shortestpaths from p to f p.

4.2 Distance computation

Consider the ellipsoid shown in the sequence in Fig. 16,where the reference point p has been placed on its back side.The focal curve f p of p is depicted in red and a point q(blue) is placed somewhere within the region bordered byf p. Our goal is to determine the distance dM (p, q), which isaccomplished by applying geodesic bifurcation.

First of all we easily obtain the black geodesic γ con-necting p and q, shown in (a), using the HA as discussedin [34]. Now we extend γ until it generates the yellow focalpoint as indicated in (b). Applying geodesic bifurcation toγ yields a family of geodesics connecting p to points onγ . We distinguish between the geodesics tracing γ upwardscolored in green respectively those tracing γ downwards,

Fig. 16 Using geodesic bifurcation to compute distances on an ellip-soid

colored in purple, see (c). The green family reflects at f p, cf.(d), where the depicted geodesic generates a focal point. Thetracing continues downwards γ until we reach the configura-tion depicted in (e) where the green geodesic finally connectsp and q. The family of purple geodesics analogously tracesdownwards γ yielding a first connecting geodesic as depictedin (f) before it reflects at f p, see (g), and ends up in the config-uration shown in (h) providing a fourth connecting geodesic.The geodesic bifurcation has yielded four connecting geo-desics from p to q as shown in the final figure (i), includingthe shortest path from p to q.

Although we exemplified the idea of computing the dis-tance on the ellipsoid in a particular example, the methoddescribed above applies to any configuration of p and q,where these points are separated by f p. Thus the outlinedapproach always yields four (distance-) relevant connect-ing geodesics, meaning that one of them is guaranteed tobe the (not necessarily unique) shortest path from p to q.This is illustrated in Fig. 17 where the symmetry set (light

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Geodesic bifurcation on smooth surfaces

Fig. 17 Distance relevantconnecting geodesics

red) decomposes the region bordered by the focal curve intofour sub-regions, see also [8]. The figure shows the result-ing set of distance relevant geodesics for a point in each ofthose sub-regions, with the shortest path being marked incyan. The blue geodesics all intersect the cut locus, being thelonger vertical branch of the symmetry set.

Having discussed the most involved case, we can state, thatthe distance computation problem for all configurations of pand q is easily achieved by computing an initial connectinggeodesic γ and applying geodesic bifurcation adequately.

Solving the distance problem using the presented approachis feasible in this case due to the fact that the consideredellipsoid has a single closed focal curve, implying that everygeodesic generates a focal point and can therefore be used toapply geodesic bifurcation. We expect this method to gener-alize to similar surfaces with a closed focal curve as exempli-fied in Fig. 18, showing a distance-relevant set of connectinggeodesics that has been obtained by applying geodesic bifur-cation to the black geodesic.

4.3 Natural starting points

Having discussed elementary distance computations we turnour attention to finding natural starting points for a redun-dancy minimizing computation of cut loci. Please recall thatthe cut locus has a graph structure consisting of cut locusbranches, which are subsets of corresponding medials char-acterized by the medial Eq. (3). In order to trace those medialswe differentiate (3) with respect to λ and obtain the medialdifferential equation (MDE)

Fig. 18 Closed focal curve on a topological sphere

∂Op

∂ss + ∂Oq

∂ϕϕ = ∂Oq

∂ss + ∂Oq

∂ϕψ, (12)

where the partial derivatives of Op on the left are evaluated in(s, ϕ), whereas on the right the corresponding parameters aregiven as (s, ψ). Obviously a starting respectively end pointon the medial is required to initiate respectively terminatethe tracing process which can be performed using standardnumerical methods, see e.g. [2,7].

Each cut locus branch has the topological structure ofan interval which may be unbounded in one or both direc-tions. If it terminates, the corresponding end point is eithera focal cusp or circumcenter as described in Sect. 2. There-fore, in order to minimize redundant computations, we startthe aforementioned tracing process in these terminal pointsif they exist and are known. If such points fail to exist asfor example in some Voronoi diagram of two points p, q,we start tracing at the middle point of a distance minimizinggeodesic connecting p with q. We will discuss each of thosethree natural starting points in the following three subsec-tions.

4.3.1 Focal cusp

We will now consider cut locus branches which are locallyinduced. Here focal cusps serve as natural starting pointsfor tracing these branches. However, as the MDE becomessingular in the focal cusp, an initial tangent of the sym-metry set has to be obtained in a different manner. RecallFig. 19 illustrating the behavior of the MDE near the focalcusp.

Considering Eq. (10) describing how to deal with the geo-desic bifurcation there. It implies a symmetrical situationfor the two appearing geodesics, as indicated by the greenparabola shown in Fig. 13 on the right. This parabola is col-ored in black in the numerical example depicted in Fig. 19 onthe right, approximating the light red preimage of the sym-metry set under Op.

Therefore taking a small Euler step using this tangentinformation in both directions yields two new geodesics(blue) providing suitable initial values for the MDE. Afterthis step, one is able to continue a regular tracing process

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Fig. 19 Initial geodesics for the MDE and approximation (black) ofthe symmetry set (light red) in parameter space

as described by the MDE. Thus, the geodesic bifurcationgives access to the cut locus branch from its natural start-ing point in the focal cusp, as confirmed by the numericalexample.

4.3.2 Middle point

As our second case we consider determining the cut locusC{p,q} of two points p, q ∈ M for which no a-priori infor-mation about potential initial respectively terminal points isavailable. In order to obtain a natural starting point for thisglobally induced medial we determine a distance minimizingconnecting geodesic and its mid point as illustrated in Fig.20a (left). From this point we start the tracing process in bothdirections as shown in the other two pictures. Note that thereappears to be a reflection behaviour of the considered medialthat is examined briefly in the following.

Using the terminology of 3.1 we can phrase the tracingprocess in terms of two geodesic families tracing the medialbranch. More precisely taking scalar products of equation(12) with ∂Op

∂s respectively ∂Oq∂s yields

(1 −

⟨∂Op

∂s,∂Oq

∂s

⟩)s =

⟨∂Oq∂ϕ,∂Op∂s

⟩ψ

(1 −

⟨∂Oq

∂s,∂Op

∂s

⟩)s =

⟨∂Oq∂ϕ,∂Oq∂s

⟩ϕ.

By subtraction we obtain⟨∂Oq

∂ϕ,∂Oq

∂s

⟩ϕ =

⟨∂Oq

∂ϕ,∂Op

∂s

⟩ψ,

indicating ψ to vanish as soon as the tracing process of themedial branch passes a focal point with respect to p, where∂Oq∂ϕ

= 0. Vice versa we have that ϕ vanishes if ∂Oq∂ϕ

= 0. The

vanishing of ψ indicates that the family of geodesics startingin p is reflected, which is immediate when incorporating theresults of Sect. 3.1. Since the two families are coupled via Eq.(12) the endpoints of the family starting in q has to exhibit asimilar behavior.

In our example the reflection of the family starting in pcauses the medial to exhibit irregular points precisely at thefocal curve f p (dark red) as confirmed in Fig. 20b (left).The point of self-intersection is actually located on the cutlocus C p which is contained in C{p,q}. After the considera-tions already discussed, the astute reader will realize that thisimplies the existence of another medial being locally inducedwith respect to p and terminating in the focal cusp of f p, seeFig. 20b (middle). In order to obtain the Voronoi diagramC{p,q} these two medials have to be cut adequately as shownin Fig. 20b (right), emphasizing that several segments of thetraced medials are in fact redundant.

Following up on the remark in 2.4 we note that the result-ing geodesic Voronoi diagram exhibits a peculiar behaviournot usually observed due to the ordinary definition of Voronoidiagrams relying on Voronoi regions partitioning the under-lying space with respect to the distance function. The gen-eralized definition based on the cut locus encodes additionaldistance information as indicated by the lower branch shown

Fig. 20 Two-point Voronoi diagram in the presence of focal curves. a Tracing a medial with reflecting behaviour, starting from the midpoint of aconnecting geodesic. b Symmetry set consisting of two medials, with the appropriate segments yielding the cut locus

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Geodesic bifurcation on smooth surfaces

Fig. 21 Example for a circumcircle homotopy

in Fig. 20b (right) which is contained in the Voronoi regionVp associated with p. More precisely any geodesic connect-ing p with another point inside Vp is distance minimizingprecisely when it does not intersect the mentioned branch.

It is now evident that incorporating the focal informationwith respect to p yields in fact two natural starting points,being the branching point b respectively the focal cusp c.The latter is locally accessible using the geodesic bifurcationas explained in Sect. 4.3.1 and should be preferred to startthe complete cut locus tracing process, while it should termi-nate in c. The problem of detecting branching points duringthe tracing process leads us to the considerations in the nextsubsection.

4.3.3 Circumcenter

In order to detect a branching point we rely on the circum-circle characterization discussed in Sect. 2. As an exampleconsider the three black sites illustrated in Fig. 21, wherethe point highlighted in red is the single branching point intheir Voronoi diagram and therefore a natural starting point.We construct a family Gλ of geodesic circles with centersm(λ) = Op1(s(λ), ϕ1(λ)) and radius s(λ) which ends up inthe desired circumcircle shown in the right picture by con-sidering the equations

Op1(s(λ), ϕ1(λ)) = Op2(s(λ), ϕ2(λ))

Op3(s3(λ), ϕ3(λ)) = Op1(s(λ), ϕ1(λ)), (13)

where the first describes the medial between p1 and p2,whereas the second specifies a geodesic connecting p3 andthe current point on the aforementioned medial. Differenti-ating this system with respect to λ one obtains a differentialequation which combines the MDE and the HA to trace thefamily Gλ of circles, terminating in the desired circumcircleif s3(λ) ≥ s(λ).

The tracing process starts with λ = 0 in the situation asdepicted in Fig. 21 (middle). In order to obtain the corre-sponding initial parameters we determine the shortest con-necting geodesic (blue) of p1 and p2, the corresponding mid-dle point m(0) and additionally the shortest connecting geo-

Fig. 22 Sequence of the tracing process of a geodesic Voronoi diagramwith six sites

desic from p3 to m(0) (not shown) with s3(0) ≥ s(0). Weend up in the configuration illustrated in the right picturewhere the three involved geodesics (blue) are of equal lengthand thereby end up on the center m(λ1) of their circumcircle,which is the single branching point in their Voronoi diagram.

In the three site example obviously p3 is the only siterelevant to be considered for inducing a branching point ofthe Voronoi regions associated with p1, p2, p3. However inthe presence of further sites p4, . . . , pn one easily extendsthe above method to detect when the medial of p1 and p2

enters the Voronoi region of any other site. For this purposeone merely has to append the additional equations

Opk (sk(λ), ϕk(λ)) = Op1(s(λ), ϕ1(λ)) for k = 4, . . . , n.

to the system (13) and proceed analogously, sopping if s(λ)becomes larger than any sk(λ) for k = 3, . . . , n.

We conclude by noting that although we used our meth-ods mainly to discuss and understand the singularities of theexponential map in terms of the reflection behavior of thegeodesic families involved, the circumcircle approach com-bining those methods with natural starting points as discussedin this paper avoids redundant tracing of medials. This is illus-trated in our example depicting a geodesic Voronoi diagramof six sites on a curved surface with inherent focal curves ofthe reference sites shown in Fig. 22.

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Fig. 23 Example for symmetrysets (upper row) andcorresponding Voronoi diagramscomputed on the manifold givenby (14) with μ = 3.0, 2.9, 2.3

4.4 Concluding example

Note that our methods are able to deal with highly non linearand curved manifolds with site constellations defining geo-desic Voronoi diagrams whose topological structure dependsinherently on the underlying Riemannian metric as indicatedby our final example depicted in Fig. 23. Here a one parame-ter family of surfaces parametrized by

f (u, v) = (u, v, u2 + v2 + μ(cos(u)+ sin(v))) (14)

with a perturbation parameter μ. To facilitate the visualiza-tion we have employed a stereographic projection diminish-ing the distracting distortions.

Comparing to our previous example in Fig. 20b (right)which might have raised the impression of being patholog-ical in the sense that the observed additional branch mightbe a rare phenomenon, our final example shows that thisbehaviour generically occurs together with sudden topolog-ical changes as μ varies. More specifically for μ = 3.0 thepicture on the left topologically reminds of an EuclideanVoronoi diagram with three collinear sites. A small variationof μ produces a branching point and furthermore the appear-ance of two locally induced cut locus branches as shown inthe middle. The corresponding symmetry set depicted aboveobviously contains several redundant medial segments whichdo not have to be traced when applying the discussed meth-ods. A further variation of μ leads to the appearance of aclosed Voronoi region with two branching points in the con-sidered region. Remarkably, those two branching points arecenters of two distinct circumcircles being described by thesame three sites.

Note that our computational approach inspired by thesmooth paradigm of classical Riemannian geometry allowsto analyze and understand these commonly occurring multi-farious phenomena.

5 Conclusion

In this follow-up on [34] we presented a solution to dealwith the singularity occurring within the homotopy basedapproach proposed there. More precisely we have analyzedsituations, where the homotopy curve meets the focal curvetangentially in a regular focal point and even in the focalcusp. A closer examination leads us to study the bifurcationof two transversally intersecting solution branches in parame-ter space. We have analytically investigated this bifurcationand exposed tangent and higher order information of thesebranches in terms of locally available information. Further-more we have shown how to exploit this information fora numerically stable implementation within the homotopyapproach.

Applications such as the medial axis or the cut locus,though being global concepts are related to the local conceptof focal curves as classical theory predicts their branches tooriginate in the examined focal cusps. The computation ofthese concepts is efficiently and accurately performed trac-ing medial branches using the medial differential equation.Unfortunately the latter becomes singular in the origins ofthe medial branches. Having solved the geodesic bifurcationin the focal cusp we are able to provide two initial geodesicslying on the corresponding medial branch. Thereby using thisresult it is possible to naturally approach the medial branchfrom the focal cusp.

The geodesic bifurcation presented in this paper allowsgeodesics generating a focal point to be used as natural homo-topy curves, dispensing the homotopy method in [34] fromany restrictions. Furthermore in case of the commonly occur-ring locally induced non-injectivities of the exponential map,such a geodesic is always available and can be used to obtaina distance relevant set of connecting geodesics incorporatingthe HA adequately.

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Especially for the illustrated surfaces with a single closedfocal curve, where every geodesic generates a focal point, thisis a valuable contribution. In this case, as we have discussed,the method is able to calculate the distance of arbitrary points.Contrary to previous approaches, this is done without requir-ing the explicit knowledge of the focal curve and withoutrelying on the artificial construction of a homotopy curve.Thus, by incorporating the geodesic bifurcation presented inthis paper, the homotopy approach unfolds its full potentialin the context of distance calculation.

Furthermore we have shown how to apply the presentedconcepts for the computation of cut loci exploiting naturalstarting points using the geodesic bifurcation. By combiningthe HA and the medial differential equation we were able tocapture the branching points of Voronoi diagrams. Thus, wewere able to minimize redundant computations of segmentsof medials.

Although the geodesic bifurcation significantly improvesupon the homotopy approach, it is limited since the gen-eral non-injectivities of the exponential map cannot com-pletely be understood considering focal situations only. Ingeneral we can distinguish between two different origins forthe appearance of multiple geodesic connections. One canbe considered to be induced by Gaussian curvature in thepresence of focal curves, whereas the other case is inducedby the global topology of the surface. Having covered thelocal aspects of geodesic ambiguity, the global aspects ofgeodesic ambiguity remain theoretically and computation-ally challenging. The latter requires additional tools provid-ing topologically distinct geodesic loops. This can be con-sidered a large topic for future research.

While being beyond the scope of this work, please notethat aside from the geodesic polar coordinates consideredhere our approach applies directly to the more general Fermicoordinates, too. In addition, due to the intrinsic setting onlyrelying on the metric tensor, our methods apply to energyminimization as indicated by Maupertuis’ principle. Further-more, our method generalizes to pseudo-Riemannian spaceswhich can be investigated in an analogous way.

Acknowledgments This research was partially supported by aDeutsche Forschungsgemeinschaft (DFG) Grant within theGraduiertenkolleg 615.

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Hannes Thielhelm obtained adiploma degree in Mathematicsand Computer Science from theLeibniz University of Hannoverin 2007. As a Ph.D. student he iscurrently involved with theoreti-cal investigations and the designof algorithms for distance relatedproblems in Riemannian geome-try.

Alexander Vais gained a B.Sc.and M.Sc. degree in ComputerScience from the Leibniz Univer-sity of Hannover (LUH) in 2007and 2009, respectively. He is cur-rently a Ph.D. student at LUHDivision of Computer Graphics.His main interests are computa-tional differential geometry andgeometry processing algorithms.

Franz-Erich Wolter has beenChaired Full Professor of Com-puter Science at Leibniz Univer-sity Hannover since 1994 wherehe directs the Division of Com-puter Graphics called Welfen-lab. He held faculty positions atUniversity of Hamburg (1994),MIT (1989–1994) and PurdueUniversity USA (1987–1989).He was software and develop-ment engineer with AEG (Ger-many) (1986–1987). Dr. Wolterobtained his Ph.D. (1985) math-ematics, TU Berlin, Germany, in

the area of Riemannian manifolds, Diploma (1980), FU Berlin, mathe-matics and theoretical physics. At MIT he co-developed the geometricmodeling system Praxiteles for the US Navy and published papers thatbroke new ground applying concepts from differential geometry andtopology on problems in geometric modeling. Dr.Wolter is researchaffiliate of MIT.

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